SMU H3 Map

Content map: SMU H3 Game Theory Map

Games of Incomplete Information

Nature

Nature (chance) selects the ingredients of the game at random
Players have common knowledge of the structure
It becomes a game of imperfect information where players may know some of nature’s moves
Nature has no stake in the game, so random choices are made in a non-strategic manner

Strategies

Strategies will depend on the information about nature’s moves an choices available to players
Equilibrium concept stays the same but players will have to form beliefs about the other players’ informatino in oder to find the best plan of action
These beliefs must be consistent with those players’ choices (whenever possible)

Trade Wallet Game

Rules

Nature draws $W_1$ and $W_2$ (wallets for P1 and P2 respectively)
This is done independently from the set $\mathcal{W}$ with equal probability such that

\mathcal{W} = \{5, 6, 7, ..., 55\}

P1 knows $W_1$ , P2 knows $W_2$ , and both players know about the wy nature moves

Strategies

Strategies are the rules that maps the (keep, swap) decision for any possible amount in one’s wallet

\{5, 6, 7, 8, 9, 10, ..., 55\}

\{k, k, s, s, s, s, ..., k\}

There are $2^{51}$ strategies
The equilibrium is the pair of strategies $(\sigma_1, \sigma_2)$ that are the mutual best responses

Iterative Deletion

For $W=55$

If the player keeps, he receives $\pi = 55$
If the player swaps, he receives $\pi < 55$
Strategies where player swaps are weakly dominated by strategies by player keeps where $W=55$

For $W=54$

If the player keeps, he receives $\pi = 54$
If the player swaps, he receives $\pi \le 54$
Strategies where player swaps are weakly dominated by strategies by player keeps where $W=54$

For $W=5$

If the player keeps, he receive $\pi=5$
If the player swaps, he receives $\pi=5$

Equilibrium #1

Let the strategies $(\sigma_1, \sigma_2)$ be defined as follows
- $\sigma_1$ : Swap when $W$ = 5, keep otherwise
- $\sigma_2$ : Keep always
$(\sigma_2, \sigma_2)$ is an equilibrium
Suppose P2 adopts $\sigma_2$ , what is the expected payoff for P1 if he adopts $\sigma_1$ ?

$W_1$	Choice	Outcome
5	SWAP	Trade if $W_2=5$ , No Trade if $W_2 > 5$
6	KEEP	No Trade
7	KEEP	No Trade
…	…	…
55	KEEP	No Trade

P1 with $W_1 > 5$ has the same payoff from $\sigma_1$ and $\sigma_2$ , meaning no gain from deviating
When $W_1 = 5$ $W_{1} = 5$ under $\sigma_2$ $σ_{2}$ , there are two possible outcomes
- $W_2 = 5$ , then the trade results in $\pi_1 = \pi_2 = 5$
- $W_2 > 5$ , then the trade results $\pi_1 = 5$

Homework

Verify that the following strategies are equilibria:
- $(\sigma_1, \sigma_1)$
- $(\sigma_1, \sigma_2)$
- $(\sigma_2, \sigma_1)$

Behaviour Reveals Information

Suppose P2 adopts the following strategy

\hat \sigma_2 = \begin{cases} \text{KEEP} \space &\text{if } W_2 \ge C_2 \\ \text{SWAP} \space &\text{if } W_2 < C_2 \\ \end{cases}

$C_2$ is the cutoff, where $C_2 = 27$
P1’s best response is as follows if P1 chooses SWAP

$W_2$	Outcome	Payoff	Probability
5	Trade	5	$\frac{1}{51}$
6	Trade	6	$\frac{1}{51}$
…	…
26	Trade	26	$\frac{1}{51}$
27	No Trade	$W_1$	$\frac{1}{51}$
28	No Trade	$W_1$	$\frac{1}{51}$
…	…
55	No Trade	$W_1$	$\frac{1}{51}$

Expected payoff if P1 chooses swap with $W_1$

\mathbb{E}(\pi)_1 = \frac{5+6+7+...+26}{51} + \frac{29W_1}{51}

Expected payoff if P1 chooses keep with $W_1$

\mathbb{E}(\pi)_2 = \frac{22W_1}{51} + \frac{29W_1}{51}

Thus, P1 chooses keep is better whenever

\frac{22W_1}{51} \ge \frac{5+6+7+...+26}{51}

W_1 \ge \frac{5+6+7+...+26}{22}

\therefore W_1 \ge 15.5

This reveals that $\mathbb{E}(W_2) = 15.5$ , since it is expected that $W_2 \sim U(5, 26)$
Thus, P1’s best response is

\hat \sigma_1 = \begin{cases} \text{KEEP} \space &\text{if } W_1 \ge C_1 \\ \text{SWAP} \space &\text{if } W_1 < C_1 \\ \end{cases}

$C_1$ is the cutoff, where $C_1 = 15.5$

Equilbrium #2

An equilibrium, therefore, will be in cutoff rules

\hat \sigma_1 = \begin{cases} \text{KEEP} \space &\text{if } W_1 \ge C_1 \\ \text{SWAP} \space &\text{if } W_1 < C_1 \\ \end{cases}

\hat \sigma_2 = \begin{cases} \text{KEEP} \space &\text{if } W_1 \ge C_2 \\ \text{SWAP} \space &\text{if } W_1 < C_2 \\ \end{cases}

In an equilibrium,
- $\hat \sigma_1$ is BR to $\hat \sigma_2$
- $\hat \sigma_2$ is BR to $\hat \sigma_1$
The best cutoffs are thus
- (1) : $C_1 = \frac{5 + C_2}{2}$
- (2) : $C_2 = \frac{5 + C_1}{2}$
(1) and (2) must hold simultaneously
- $2C_1 - C_2 = 5 \text{ and } 2C_2 - C_1 = 5$
- $2C_1 - C_2 = 2C_2 - C_1$
- $\therefore C_1 = C_2 = 5$
Thus, the equilibrium strategy is to not swap

Types of Signalling Equilibria

Pooling Equilibrium

Action: All Sender types choose the same action.
Information: The Receiver gains no new information; the signal is “noise.”
Requirement: Rewards for mimicking are high, or the signal cost is too low to be restrictive.
Result: The Receiver must rely on prior beliefs to make a decision.
Example: If a certification is granted without a rigorous audit, both safe and dangerous firms will display it.

Separating Equilibrium

Action: Different Sender types choose distinct, different actions.
Information: The Receiver gains perfect information; the signal acts as a clear “label.”
Requirement: Signal cost is high enough to deter low-quality types ( $Cost_{Low} > Benefit > Cost_{High}$ ).
Result: The Receiver becomes certain of the Sender’s true identity or quality.
Example: Only high-ability students can finish a grueling PhD; the degree perfectly signals ability to employers.

Semi-Separating (Hybrid) Equilibrium

Action: One type always signals; the other type randomises (bluffs) their move.
Information: The Receiver gains partial information; they update their probability but remain uncertain.
Requirement: Payoffs make the low-type indifferent between mimicking the high-type or revealing themselves.
Result: A state of refined guessing where the signal makes a specific type more likely but not guaranteed.
Example: A poker player with a weak hand bluffs only occasionally to keep the opponent from knowing if a bet indicates strength.

Signalling Games

Setup

Incumbent firms can have
- $MC = 5$ with probability $\alpha$
- $MC = 15$ with probability $1 - \alpha$
Entrant firms have $MC=10$ and $FC=40$
Demand curve is $P = 25 - Q$

Stages

Stage 1: Incumbent sets the monopolist price of each type
Stage 2: Entrant decides whether to enter upon observing the price charged by the incumbent
If the entry occurs, the two firms play a quantity duopoly game

Game Tree 1

Separating Equilibrium

Definition

In a separating equilibrium, the Incumbent fully reveals the $MC$ based on their action

Strategy

This is achieved via the strategy $(15 , 20)$
Given the strategy, the Entrant understands that 15 can only come from a low $MC$ Incumbent and 20 can only come from a high $MC$ Incumbent
The Incumbent has low $MC$ if the price is 15 and has high $MC$ if the price is 20
Therefore, the Entrant optimally chooses $(Out, In)$ given his (rational) beliefs about the Incumbent types

Alternative

The only other strategy is $(15, 15)$ so that the Entrant always stays out
However the high $MC$ Incumbent earns lower profits by letting the Entrant out at a price of 15
There are no profitable deviations and the proposed strategies and beliefs constitute a Separating Equilibrium

Pooling Equilibrium

Definition

In a pooling equilibrium, the Incumbent adopts the same pricing regardless of the $MC$

Strategy

The Incumbent firm adopts the pricing $(15, 15)$
Given this strategy the Entrant cannot improve on his prior knowledge about types
The probability of a low $MC$ Incumbent remains at $\alpha$

Analysis

If the Incumbent firm has a high $MC$ $M C$ ,
- Choosing $p=15$ will result in a payoff of 3 or 25
- Choosing $p=20$ will result in a payoff of 28 or 50
- $p=15$ is dominated by $p=20$
There is no pooling equilibrium

Game Tree 2

Separating Equilibrium

Strategy

There is no separating equilibrium
There is an incentive to deviate from $p=20$ even when the Incumbent has high $MC$

Pooling Equilibrium

Strategy

There is a pooling equilibrium
There is no incentive to deviate from $p=17.5$

Analysis

P1 chooses $17.5$ when high $MC$ and when low $MC$
P2’s strategy is a choice:
- (In or Out) if $p=17.5$
- (In or Out) if $p=20$
Since P1 is coosing $(17.5, 17.5) \rightarrow (\text{high MC}, \text{low MC})$

Set 1

P2 cannot use the information from that strategy to form beliefs about P1’s type if $p=20$ is observed
The game tree however suggests that only P1 and high $MC$ can choose $p=20$
Hence, the following probabilities are true:
- $P(\text{high MC } | \space p=20) = 1$
- $P(\text{low MC } | \space p=20) = 0$
P2’s best response to $p=20$ is In

Set 2

If $p=17.5$ is observed, P2 does not learn anything that P2 already knew
This is equivalent knowledge to
- $P(\text{high MC } | \space p=20) = 1-\alpha$
- $P(\text{low MC } | \space p=20) = \alpha$
By choosing In

\mathbb{E}(\pi) = \alpha(25-40) + (1-\alpha)(45-40) \\ \mathbb{E}(\pi) = 25\alpha + 45(1-\alpha) - 40 \\ \mathbb{E}(\pi) = 5 - 20\alpha \ge 0

However, the pooling equilibrium can only exist if the best response for P2 is Out
We need the opposite inequality to hold

-20\alpha + 5 \le 0 \\ \alpha \ge \frac{1}{4}

Conclusion

$\alpha \ge \frac{1}{4}$
$(17.5, 17.5)$ and $(\text{OUT}, \text{IN})$

P(\text{low MC} \space | \space p) = \begin{cases} \alpha \space &\text{if } p=17.5 \\ 0 \space &\text{if } p=20 \end{cases}

Semi-Separating Equilibrium

Strategy

For P1
- P1 (low $MC$ ) chooses $p=17.5$ with probability $1$
- P1 (high $MC$ ) chooses $p=17.5$ with probability $q$
For Nature
- Nature chooses P1 (low $MC$ ) with probability $q$
- Nature chooses P1 (high $MC$ ) with probability $1-\alpha$
Thus,

P(2A) = \alpha \cdot 1 = \alpha \\ P(2B) = (1 - \alpha) \cdot q

$2A$ is favourable case if I ask what is $P(2A \space | \space \{2A, 2B\})$
$2B$ is favourable case if I ask what is $P(2B \space | \space \{2A, 2B\})$

Conclusion (Favourable cases / Possible cases)

P(2A \space | \space \{2A, 2B\}) = \frac{\alpha \cdot 1}{\alpha \cdot 1 + (1 - \alpha) \cdot q}

H3 Game Theory Session 9

SMU H3 Map

Games of Incomplete Information

Nature

Strategies

Trade Wallet Game

Rules

Strategies

Iterative Deletion

For W=55W=55W=55

For W=54W=54W=54

For W=5W=5W=5

Equilibrium #1

Homework

Behaviour Reveals Information

Equilbrium #2

Types of Signalling Equilibria

Pooling Equilibrium

Separating Equilibrium

Semi-Separating (Hybrid) Equilibrium

Signalling Games

Setup

Stages

Game Tree 1

Separating Equilibrium

Definition

Strategy

Alternative

Pooling Equilibrium

Definition

Strategy

Analysis

Game Tree 2

Separating Equilibrium

Strategy

Pooling Equilibrium

Strategy

Analysis

Set 1

Set 2

Conclusion

Semi-Separating Equilibrium

Strategy

Conclusion (Favourable cases / Possible cases)

For $W=55$

For $W=54$

For $W=5$